Open research questions in Point processes and geometric inequalities

What the literature leaves open

• The non-convexity measure considered is unstable in the Hausdorff metric. There is a need for a modified non-convexity measure that is stable for convex sets.

  On the numerical calculation of the angular measure of non-convexity of alpha-sets · 2026 · DOI

• The Lp Minkowski problem remains unsolved for p < 1. The problem of characterizing geometric measures of convex bodies is fundamental and remains open.

  The Lp dual Minkowski problem for group-invariant convex bodies · 2026 · DOI

• The problem of finding a capillary convex body with a prescribed k-th capillary area measure is a challenging problem. The authors need to establish the existence and uniqueness of a smooth solution under a natural sufficient condition.

  The capillary Christoffel–Minkowski problem · 2026 · DOI

• The capillary Christoffel-Minkowski problem is a new problem that has not been studied before. The authors identify a gap in the literature on the study of capillary convex bodies.

  The capillary Christoffel–Minkowski problem · 2026 · DOI

• Detecting gaps in the lattice point distribution. Developing algorithms for computing the arithmetic width. Establishing connections between the arithmetic width and discrete geometry, integer programming, and additive combinatorics.

  An arithmetic measure of width for convex bodies · 2026 · DOI

• To further study the asymptotic density of iterated lattice sumsets in higher dimensions. To explore applications of the universal additive self-healing property in coding theory and cryptography.

  Asymptotic Density of Iterated Lattice Sumsets and Additive Self-Healing in Rectangular Lattice Boxes (Version 1.12) · 2026 · DOI

• The lack of a comprehensive understanding of the asymptotic density of iterated lattice sumsets. The need for a new classification of lattice sets by their asymptotic sumset-filling behavior.

  Asymptotic Density of Iterated Lattice Sumsets and Additive Self-Healing in Rectangular Lattice Boxes (Version 1.12) · 2026 · DOI

• The literature lacks a proof of the Wulff inequality for minimal submanifolds with boundary. The paper fills this gap by proving the inequality using a new technique.

  Wulff inequality for minimal submanifolds in Euclidean space · 2026 · DOI

• Future research can focus on applying the results of this paper to various areas. The study of rearrangements and their approximation by polarizations can be further developed and generalized.

  Approximation of rearrangements by polarizations · 2026 · DOI

• The paper identifies a gap in the study of rearrangements and their approximation by polarizations. The authors aim to fill this gap by developing the theory of rearrangements and their associated set maps.

  Approximation of rearrangements by polarizations · 2026 · DOI

• The method assumes that the bodies are axially symmetric. The method does not provide sufficient conditions for relative equilibria. The method is limited to the case of axially symmetric bodies.

  A general framework for relative equilibria in the symmetric full gravitational N-Body problem · 2026 · DOI

• To extend the method to the case of triaxial bodies. To study the stability of relative equilibria using the method. To apply the method to other problems in astrophysics and celestial mechanics.

  A general framework for relative equilibria in the symmetric full gravitational N-Body problem · 2026 · DOI

• To study the minimal area problem for polynomial lemniscates with different types of compact sets. To determine the exact decay rate of the minimal area for specific compact sets.

  A note on the Erdős minimal area problem · 2026 · DOI

• The minimal area problem for polynomial lemniscates was posed by Erdös, Herzog, and Piranian, but the decay rate was not known. The relationship between the logarithmic capacity of a compact set and the minimal area of its polynomial lemniscates was not understood.

  A note on the Erdős minimal area problem · 2026 · DOI

• The paper identifies a gap in the existing literature on the desingularization of real algebraic sets. The paper proposes a new method for uniformizing closed chessboard sets.

  Nash Uniformization of Chessboard Sets by Nash Manifolds with Corners · 2026 · DOI

• Further study of the Lp Minkowski problem for p < 1. Investigation of the geometry of irreducible closed subgroups in other contexts.

  The Lp dual Minkowski problem for group-invariant convex bodies · 2026 · DOI

• The case of higher dimensions could be studied. The construction of Steinhaus could be further generalized.

  Buffon discrepancy and the Steinhaus longimeter · 2026 · DOI

• A major obstacle is that the existing technique for arbitrarily-sized squares uses only axis-aligned cuts, while even for disks, axis-aligned cuts alone are insufficient to obtain a separable subset of linear size.

  On Linear-Size Guillotine-Separable Subsets of Fat Convex Objects, Disks, and Squares · 2026

• We resolve this longstanding open problem by proving that every family of pairwise disjoint fat convex objects has a separable subset of linear size.

  On Linear-Size Guillotine-Separable Subsets of Fat Convex Objects, Disks, and Squares · 2026

• This restriction represents a notable gap in the literature as variables with mixed sign domains arise naturally in practice.

  Volume Formulae for the Convex Hull of the Graph of a Trilinear Monomial: A Complete Characterization for General Box Domains · 2026 · DOI

• We prove the Weinstock inequality for the first Steklov eigenvalue of convex domains in hyperbolic space $\mathbb{H}^{n}$, resolving Open Question 4.

  The Weinstock inequality for convex domains in hyperbolic space · 2026

• The problem of distributing a one-dimensional set with total length is not well understood. The Buffon discrepancy is not well studied.

  Buffon discrepancy and the Steinhaus longimeter · 2026 · DOI

• The paper identifies a gap in the literature on the Orlicz-Minkowski problem and its variations. The authors note that the capillary Orlicz-Minkowski problem has not been previously studied.

  The capillary Orlicz-Minkowski problem · 2026 · DOI

• Future research can focus on developing more efficient algorithms for computing the arithmetic width. The arithmetic width can be used to study the behavior of lattice points in convex bodies. The results of the paper can be generalized to other types of convex bodies and geometric objects.

  An arithmetic measure of width for convex bodies · 2026 · DOI

• The study of the torsional rigidity of multiply connected domains in higher dimensions. The development of a general result for the stability of the annulus for the torsion of multiply connected domains. The application of the results of the paper to optimize the design of beams with multiply connected cross-sections.

  On the stability of the annulus for the torsion of multiply connected domains · 2026 · DOI

Most-cited papers in Point processes and geometric inequalities

• Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case · Computational Methods in Applied Mathematics · 2022 · 3 citations
• Convexity-preserving properties of set-valued ratios of affine functions · Studia Universitatis Babes-Bolyai Matematica · 2021 · 2 citations
• RELATION BETWEEN THE COVARIOGRAM AND DISTRIBUTION FUNCTION OF THE DISTANCE BETWEEN TWO UNIFORM AND INDEPENDENT POINTS · Proceedings of the YSU A Physical and Mathematical Sciences · 2022 · 2 citations
• On the monotonicity of non-local perimeter of convex bodies · Topological Methods in Nonlinear Analysis · 2024 · 1 citations
• Topology of the Grünbaum-Hadwiger-Ramos problem for mass assignments · Topological Methods in Nonlinear Analysis · 2023 · 1 citations
• An Interior Maximum Norm Error Estimate for the Symmetric Interior Penalty Method on Planar Polygonal Domains · Computational Methods in Applied Mathematics · 2022 · 1 citations
• CONDITIONAL MOMENTS OF THE DISTANCE DISTRIBUTION TWO RANDOM POINTS IN A CONVEX DOMAIN IN $ {\mathbb{R}}^2 $ · Proceedings of the YSU A Physical and Mathematical Sciences · 2020 · 1 citations
• A Generalization of the Finsler–Hadwiger Theorem · Mathematics Magazine · 2024 · 1 citations
• Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions · Japan Journal of Industrial and Applied Mathematics · 2026 · 0 citations
• On stochastic forms of functional isoperimetric inequalities · Revista Matemática Complutense · 2026 · 0 citations

Most recent work

• Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions · Japan Journal of Industrial and Applied Mathematics · 2026
• On stochastic forms of functional isoperimetric inequalities · Revista Matemática Complutense · 2026
• Dimension-Independent Finiteness of Central Configurations for Positive Masses · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Explicit lower bounds for opaque sets of unit square and unit disc · Acta Mathematica Academiae Scientiarum Hungaricae · 2026
• Simultaneous rotation of infinitely many parallel line segments · Proceedings of the Royal Society of Edinburgh: Section A Mathematics · 2026
• On the numerical calculation of the angular measure of non-convexity of alpha-sets · Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta · 2026
• The Lp dual Minkowski problem for group-invariant convex bodies · Advances in Mathematics · 2026
• The capillary Christoffel–Minkowski problem · Calculus of Variations and Partial Differential Equations · 2026
• The capillary Orlicz-Minkowski problem · Journal of Functional Analysis · 2026
• k-Convex Polyominoes by Semi-perimeter · Electronic Proceedings in Theoretical Computer Science · 2026

Related topics in Mathematics